Rethinking Pythagoras - 5. Summary

The ideas in the previous sections weave together several themes. Differential and integral calculus, Taylor series, Euclidean, spherical, and hyperbolic geometries, and measurement of length and area are all explicitly discussed, but straightedge and compass construction, discovery via dynamic software, and the difference between discovery and proof are also implicit in the discussion. Accordingly, the author has used the discovery activity given in the appendix during three of the last four days of the College Geometry class he teaches (the last day is alloted for review). That activity leads students to discover and prove some relationships between radius and circumference and area of circles in three geometries, and to discover (without proof) Pythagorean Theorem analogs in the non-Euclidean geometries.

There are several reasons why this activity is useful. First, all students can at least begin the activity, constructing and exploring figures in the dynamic software. While stronger students usually finish the entire activity in the alloted three class periods, most weaker students only complete the Euclidean and spherical parts, and need quite a bit of prompting from their instructor on some of the deeper questions. All of the author's students have needed assistance with step 2h, which utilizes the identity: \(A(S)=2\pi\cos(r)-2\pi\), implying that \(\cos(r)=[2\pi-A(S)]/(2\pi)\).

In fact weaker students have been unusually engaged in the activity, and often related "aha!" moments having to do with a new and improved understanding of calculus (rather than geometry). For example, when integrating \(2\pi\int\sin(r)dr\) in order to find the area of a spherical circle, one student at first complained that "the" solution, \(-2\pi\cos(r)\), didn't work. However, when he discovered that addition of the appropriate constant gave a workable solution, he was ecstatic: "That's why we needed that constant back in Calc 2! I never understood that before!"

Stronger students, on the other hand, have usually expressed "aha!" moments having to do with their understanding of geometry. One student, after discovering the spherical analog to the Pythagorean Theorem, wondered why we use the Pythagorean Theorem at all, since we live on a sphere and should be using the spherical version. After working through step 2j in the appendix, she realized why we use the simplified Euclidean Pythagorean Theorem: "Oh, so it really doesn't matter which one we use unless we either need a lot of accuracy or are traveling a long distance." A few (juniors) have been inspired to undertake senior capstone projects related to non-Euclidean geometry in their subsequent year.

Secondly, the activity gives students an opportunity to use skills from prerequisite classes in a new context. We already mentioned how weaker students have often expressed increased understanding of calculus through this activity. Even several stronger students struggled to remember calculus concepts that were becoming rusty with disuse.

A common coffee-shop topic of discussion at math conferences is students' difficulty in applying knowledge in new contexts. It makes sense to create opportunities for students to practice applications, such as by using MacLaurin series to explore the spherical Pythagorean Theorem analog in their geometry class.

Thirdly, the activities are designed to force students to the conclusion that numeric explorations can be misleading. For example, the relationship between radius and circumference of a circle in spherical geometry is explored numerically beginning on a very tiny scale. Students conclude, incorrectly, that \(C=2\pi r\), and then explore larger circles, discovering that their original conclusion was incorrect. This is a point that the instructor can use quite effectively to point out the weakness of numeric explorations and the importance of proof; if the students had stopped their explorations after looking at only small-scale circles, they would probably remain convinced that circles in spherical geometry have the same area and circumference formulas as their Euclidean counterparts.

In fact, the author has noticed that his students often are more convinced by numerical explorations than by rigorous proofs. After trying many methods to convince students of the need for proof, he finally saw the obvious: "If they are more convinced by software than by proof, then I can convince them of the need to go beyond the software by making use of their trust in the software." An exploration that leads students to create a conjecture that later turns out to be demonstrably false is one way of doing this.

Lastly, students find these numerical explorations fun. Many students are clearly more engaged in the class material while doing this kind of exploration than with any other method of instruction. While this may be because of the superficial resemblance of exploration using dynamic software to playing video games, it can nonetheless be used to encourage students to immerse themselves in geometric concepts. In particular, the author has found that the weakest students usually have the most to gain from this type of exploration.

Perhaps the most amusing evidence of the effectiveness of this activity came from this year's final exam. On it, I included the following problem, which was intended to be a simple "warm up" type problem, and at the same time to address the fact that most of my students plan to teach high school math.

The following problem comes directly from a sample problem found at yourteacher.com [21], and was meant for a high school algebra class. "Raul is 6 feet tall, and notices that his shadow is 5 feet long. The shadow of his school building is 30 feet long. How tall is his school building?"

In fact, the problem assumes knowledge of geometric ideas. Solve this problem, explaining at which steps you are using algebra and at which steps you are using geometry.

One student gave the following response, with a sketch of two appropriate triangles (parenthetical clarification added).

We know angles with the ground and [angles with the] sun are the same. Since AA [two angles are congruent], the triangles are similar. The similar triangles theorem says 6/5 = x/30, so x=36. The building is 36 feet. Except we live on a sphere, so we'd have to use a spherical similar triangles theorem, which is probably different. But since we don't care about hundredths of an inch, I guess Euclidean rules will give a good approximation. 36 feet is about right.